Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(11,13)(12,15)(17,18,21,23)(19,20,22,24);(1,2,6,4)(3,7,5,8)(9,10,14,16)(11,15,13,12)(17,18)(19,24)(20,22)(21,23)(26,29)(30,32);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,16,12)(17,19,21,22)(18,20,23,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24);(1,5,6,3)(2,8,4,7)(9,13,14,11)(10,12,16,15)(17,20)(18,22)(19,23)(21,24)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fg,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1208809);